The Toda system and multiple-end solutions of autonomous planar elliptic problems

We prove the existence of a new class of entire, positive solutions for the classical elliptic problem Δu-u+up=0 in R², when p>2. The solutions we construct are obtained by perturbing the function ∑_j=1^kw(dist({dot operator},γj)), where k≥1, w is the unique even, positive, non-constant solution of w^″-w+w^p=0 in R and where the curves γ_j are the graphs of the functions f₁,...,f_k which are solutions of the Toda system. c²f_j^″=e^fj-1-fj-e^fj-fj+1 with f₀≡-∞ and f_k+1≡+∞ This result provides a surprising link between the solutions of the Toda system and entire solutions of the above semilinear elliptic equation.